194 Mr. R. Hargreaves on a Diffraction Problem 



where 



p 1 1.3,5 1 n 1_ 1^3 jL 



r «-~2/?l + 2^2 / ol + '--' ^ a ~ pt~~2.2pi ■ ••• , V*' 



series ultimately divergent for any value of p. For p great 

 evidently 



i °° cos udu i, / \ /a / \ • ^ 



1 —=- = - Y a (p) cos p - Q (/>) sin p, ! 



( Sm /- ■ - = ~ P «0>) Sil1 /> + QaW C0S P , 



The pairs (PQ) and (P a Q a ) both satisfy equations 



y. (6 6) 



whence 





The nature of the wave expressed by <£(r, y) is revealed 

 more intimately by the use of (6 a), giving as connected 

 forms 



4(r> y) 



= 2~-|)P(^) cos kp + Q(kp) sin kp\ cos J j + k(Vt+y) L 



+ ^P(%>) sinfy>-Q(fy)cosfy>}- sin jj + £(V* + y)l"I 

 = ^ [P(W cos {j +*(V*-r)l-Q(^) 



X sin ^ --4-A;(V£ — ? 

 = 2T7"[^ P ^ C0S ^ r + Q(^) sin M cos (j + *vA 



+ {P(fy) sinkr-Q(kp) coskr\sinM +kYt\~\. 



We have here the change from plane to divergent wave, 

 and in the last line the forms of the stationary solution. 

 By use of the series P and Q we obtain solutions in 



1 



(8) 



